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New Limits on FaultTolerant Quantum Computation
"... We show that quantum circuits cannot be made faulttolerant against a depolarizing noise level of ^ ` = (62p2) /7 ss 45%, thereby improving on a previous boundof 50 % (due to Razborov [18]). More precisely, the circuit model for which we prove this bound contains perfect gatesfrom the Clifford gro ..."
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We show that quantum circuits cannot be made faulttolerant against a depolarizing noise level of ^ ` = (62p2) /7 ss 45%, thereby improving on a previous boundof 50 % (due to Razborov [18]). More precisely, the circuit model for which we prove this bound contains perfect gatesfrom the Clifford group (CNOT, Hadamard, S, X, Y, Z)and arbitrary additional onequbit gates that are subject to
Quantum Lower Bounds for Fanout
, 2003
"... We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, t ..."
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Cited by 7 (2 self)
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We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, this bound is close to optimal. In the case of a nonconstant number a of ancill and n input qubits, we give a tradeo between a and the required depth, that results in a nontrivial lower bound for fanout when a = n 1 o(1) .
The Computational Complexity of Linear Optics
 in Proceedings of STOC 2011
"... We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical n ..."
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We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomialtime classical algorithm that samples from the same probability distribution as a linearoptical network, then P #P = BPP NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the PermanentofGaussians Conjecture, which says that it is #Phard to approximate the permanent of a matrixAofindependentN (0,1)Gaussianentries, withhigh probability over A; and the Permanent AntiConcentration Conjecture, which says that Per(A)  ≥ √ n!/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. For the 96page full version, see www.scottaaronson.com/papers/optics.pdf
Quantum Algorithms for Hidden Coset Problems
, 2001
"... Many of the most successful quantum algorithms to date have exploited the ability of the Fourier transform to capture subgroup structure. However, the ability of the Fourier transform to capture shift structure has not been explored. We present three examples of "unknown shift" problems that can be ..."
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Many of the most successful quantum algorithms to date have exploited the ability of the Fourier transform to capture subgroup structure. However, the ability of the Fourier transform to capture shift structure has not been explored. We present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. By reducing the problem of breaking algebraically homomorphic cryptosystems to one of these problems, we show that such cryptosystems can be broken on a quantum computer. These shift problems and the hidden subgroup problem can be cast in the framework of the hidden coset problem. This framework gives us a unified way of viewing the quantum Fourier transform's ability to capture subgroup and shift structure.
Generalpurpose parallel simulator for quantum computing." quantph /0201042
, 2002
"... With current technologies, it seems to be very difficult to implement quantum computers with many qubits. It is therefore of importance to simulate quantum algorithms and circuits on the existing computers. However, for a largesize problem, the simulation often requires more computational power tha ..."
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With current technologies, it seems to be very difficult to implement quantum computers with many qubits. It is therefore of importance to simulate quantum algorithms and circuits on the existing computers. However, for a largesize problem, the simulation often requires more computational power than is available from sequential processing. Therefore, the simulation methods using parallel processing are required. We have developed a generalpurpose simulator for quantum computing on the parallel computer (Sun, Enterprise4500). It can deal with upto 30 qubits. We have performed Shor’s factorization and Grover’s database search by using the simulator, and we analyzed robustness of the corresponding quantum circuits in the presence of decoherence and operational errors. The corresponding results, statistics and analyses are presented. key words: quantum computer simulator, Shor’s factorization, Grover’s database search, parallel processing, decoherence and operational errors 1
www.stacsconf.org DISTINGUISHING SHORT QUANTUM COMPUTATIONS
"... Abstract. Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of ..."
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Abstract. Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for QIP on constant depth circuits containing the unbounded fanout gate. These results are shown by reducing a QIPcomplete problem to a logarithmic depth version of itself using a parallelization technique. 1.
How quantum computers can fail
"... We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system w ..."
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Cited by 4 (2 self)
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We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system with many highly correlated elements there will be a strong effect of error synchronization. These postulates appear to be damaging for quantum computers.
Detrimental Decoherence
, 2007
"... We propose and discuss two conjectures on the nature of information leaks (decoherence) for quantum computers. These conjectures, if (or when) they hold, are damaging for quantum errorcorrection as required by faulttolerant quantum computation. The first conjecture asserts that information leaks ..."
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We propose and discuss two conjectures on the nature of information leaks (decoherence) for quantum computers. These conjectures, if (or when) they hold, are damaging for quantum errorcorrection as required by faulttolerant quantum computation. The first conjecture asserts that information leaks for a pair of substantially entangled qubits are themselves substantially positively correlated. The second conjecture asserts that in a noisy quantum computer with highly entangled qubits there will be a strong effect of error synchronization. We present a more general conjecture for arbitrary noisy quantum systems: prescribing (or describing) noisy quantum systems at a state ρ is subject to error E which “tends to commute ” with every unitary operator that stabilizes ρ.
Quantum fanout is powerful
 Theory of Computing
"... We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, thr ..."
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We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fanin gates. If we allow arbitrary onequbit gates instead of a fixed basis, then these circuits can also be made exact in logstar depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth. 1